(-1.0, 4.0)
(-1.0, 4.0)
Ingeniería Biomédica
2025-03-06
Convolution is a fundamental operation in signal processing.
Given two signals ( x(t) ) and ( h(t) ), their convolution is defined as:
\[ y(t) = x(t) * h(t) = \int_{-\infty}^{\infty} x(\tau) h(t - \tau) d\tau \]
In discrete-time, convolution is:
\[ y[n] = \sum_{k=-\infty}^{\infty} x[k] h[n-k] \]
Convolution in time domain corresponds to multiplication in frequency domain:
\[ X(f) H(f) = Y(f) \]
This property is crucial in filter design and system analysis.
(-1.0, 4.0)
(-1.0, 4.0)
Fourier series represents periodic signals as a sum of sinusoids:
\[ x(t) = \sum_{n=-\infty}^{\infty} C_n e^{jn\omega_0 t} \]
where ( C_n ) are the Fourier coefficients.
Decomposing a signal into sinusoidal components allows frequency analysis.
The Fourier coefficients ( C_n ) are computed as:
\[ C_n = \frac{1}{T} \int_{0}^{T} x(t) e^{-jn\omega_0 t} dt \]
Determines how much of each frequency is present in the signal.
Sampling theorem: A signal must be sampled at a frequency at least twice its highest frequency component:
\[ f_s \geq 2 f_{max} \]
Aliasing occurs if sampling frequency is too low.